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A Formal Analysis of Onion Routing 10/26/2007 Aaron Johnson (Yale) with Joan Feigenbaum (Yale) Paul Syverson (NRL)

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Papers 1.A Model of Onion Routing with Provable Anonymity Financial Cryptography and Data Security 2007 2.A Probabilistic Analysis of Onion Routing in a Black-box Model Workshop on Privacy in the Electronic Society 2007

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Anonymous Communication Sender anonymity: Adversary cant determine the sender of a given message Receiver anonymity: Adversary cant determine the receiver of a given message Relationship anonymity: Adversary cant determine who talks to whom

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Anonymous Communication Sender anonymity: Adversary cant determine the sender of a given message Receiver anonymity: Adversary cant determine the receiver of a given message Relationship anonymity: Adversary cant determine who talks to whom

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How Onion Routing Works User u running client Internet destination d Routers running servers ud 12 3 4 5

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How Onion Routing Works ud 1.u creates 3-hop circuit through routers 12 3 4 5

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How Onion Routing Works ud 1.u creates 3-hop circuit through routers 12 3 4 5

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How Onion Routing Works ud 1.u creates 3-hop circuit through routers 12 3 4 5

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How Onion Routing Works ud 1. u creates 3-hop circuit through routers 2. u opens a stream in the circuit to d 12 3 4 5

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How Onion Routing Works ud 1. u creates 3-hop circuit through routers 2. u opens a stream in the circuit to d 3.Data is exchanged {{{m} 3 } 4 } 1 12 3 4 5

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How Onion Routing Works ud 1. u creates 3-hop circuit through routers 2. u opens a stream in the circuit to d 3.Data is exchanged {{m} 3 } 4 12 3 4 5

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How Onion Routing Works ud 1. u creates 3-hop circuit through routers 2. u opens a stream in the circuit to d 3.Data is exchanged {m} 3 12 3 4 5

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How Onion Routing Works ud 1. u creates 3-hop circuit through routers 2. u opens a stream in the circuit to d 3.Data is exchanged m 12 3 4 5

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How Onion Routing Works ud 1. u creates 3-hop circuit through routers 2. u opens a stream in the circuit to d 3.Data is exchanged m 12 3 4 5

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How Onion Routing Works ud 1. u creates 3-hop circuit through routers 2. u opens a stream in the circuit to d 3.Data is exchanged {m} 3 12 3 4 5

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How Onion Routing Works ud 1. u creates 3-hop circuit through routers 2. u opens a stream in the circuit to d 3.Data is exchanged {{m} 3 } 4 12 3 4 5

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How Onion Routing Works ud 1. u creates 3-hop circuit through routers 2. u opens a stream in the circuit to d 3.Data is exchanged {{{m} 3 } 4 } 1 12 3 4 5

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How Onion Routing Works ud 1. u creates 3-hop circuit through routers 2. u opens a stream in the circuit to d 3.Data is exchanged. 4.Stream is closed. 12 3 4 5

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How Onion Routing Works u 1. u creates 3-hop circuit through routers 2. u opens a stream in the circuit to d 3.Data is exchanged. 4.Stream is closed. 5.Circuit is changed every few minutes. 12 3 4 5 d

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Results

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1.Formally model onion routing using input/output automata

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Results 1.Formally model onion routing using input/output automata 2.Analyze relationship anonymity a.Characterize situations with possibilistic anonymity b.Bound probabilistic anonymity in worst-case and typical situations

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Related Work A Formal Treatment of Onion Routing Jan Camenisch and Anna Lysyanskaya CRYPTO 2005 A formalization of anonymity and onion routing S. Mauw, J. Verschuren, and E.P. de Vink ESORICS 2004 Towards an Analysis of Onion Routing Security P. Syverson, G. Tsudik, M. Reed, and C. Landwehr PET 2000

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Model Constructed with I/O automata –Models asynchrony –Relies on abstract properties of cryptosystem Simplified onion-routing protocol –No key distribution –No circuit teardowns –No separate destinations –Each user constructs a circuit to one destination –Circuit identifiers

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Automata Protocol u v w

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u v w

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u v w

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u v w

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u v w

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u v w

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u v w

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u v w

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u v w

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u v w

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Creating a Circuit u123

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[0,{CREATE} 1 ] 1.CREATE/CREATED u123

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Creating a Circuit [0,CREATED] 1.CREATE/CREATED u123

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Creating a Circuit 1.CREATE/CREATED u123

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Creating a Circuit 1.CREATE/CREATED 2.EXTEND/EXTENDED [0,{[EXTEND,2, {CREATE} 2 ]} 1 ] u123

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Creating a Circuit 1.CREATE/CREATED 2.EXTEND/EXTENDED [l 1,{CREATE} 2 ] u123

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Creating a Circuit 1.CREATE/CREATED 2.EXTEND/EXTENDED [l 1,CREATED] u123

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Creating a Circuit 1.CREATE/CREATED 2.EXTEND/EXTENDED [0,{EXTENDED} 1 ] u123

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Creating a Circuit 1.CREATE/CREATED 2.EXTEND/EXTENDED 3.[Repeat with layer of encryption] [0,{{[EXTEND,3, {CREATE} 3 ]} 2 } 1 ] u123

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Creating a Circuit 1.CREATE/CREATED 2.EXTEND/EXTENDED 3.[Repeat with layer of encryption] u123 [l 1,{[EXTEND,3, {CREATE} 3 ]} 2 ]

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Creating a Circuit 1.CREATE/CREATED 2.EXTEND/EXTENDED 3.[Repeat with layer of encryption] [l 2,{CREATE} 3 ] u123

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Creating a Circuit 1.CREATE/CREATED 2.EXTEND/EXTENDED 3.[Repeat with layer of encryption] [l 2,CREATED] u123

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Creating a Circuit 1.CREATE/CREATED 2.EXTEND/EXTENDED 3.[Repeat with layer of encryption] [l 1,{EXTENDED} 2 ] u123

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Creating a Circuit 1.CREATE/CREATED 2.EXTEND/EXTENDED 3.[Repeat with layer of encryption] [0,{{EXTENDED} 2 } 1 ] u123

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Adversary u 12 3 4 5 d Active & Local

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Possibilistic Anonymity

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Anonymity Let U be the set of users. Let R be the set of routers. Let l be the path length.

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Anonymity Definition (configuration): A configuration is a function U R l mapping each user to his circuit. Let U be the set of users. Let R be the set of routers. Let l be the path length.

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Anonymity Definition (indistinguishability): Executions and are indistinguishable to adversary A when his actions in are the same as in after possibly applying the following: : A permutation on the keys not held by A. : A permutation on the messages encrypted by a key not held by A. Definition (configuration): A configuration is a function U R l mapping each user to his circuit. Let U be the set of users. Let R be the set of routers. Let l be the path length.

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Definition (relationship anonymity): u and d have relationship anonymity in configuration C with respect to adversary A if, for every fair, cryptographic execution of C in which u talks to d, there exists a fair, cryptographic execution that is indistinguishable to A in which u does not talk to d. Anonymity Definition (fair): In fair executions actions enabled infinitely often occur infinitely often Definition (cryptographic): In cryptographic executions no encrypted control messages are sent before they are received unless the sender possesses the key

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Theorem 1: Let C and D be configurations for which there exists a permutation : U U such that C i (u) = D i ( (u)) if C i (u) or D i ( (u)) is compromised or is adjacent to a compromised router. Then for every fair, cryptographic execution of C there exists an indistinguishable, fair, cryptographic execution of D. The converse also holds.

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C u v 12 3 4 5

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u v 12 3 4 5 3 2 CD

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u v 12 3 4 5 3 2 CD v u 225 4

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u v 12 3 4 5 CD Theorem 1: Let C and D be configurations for which there exists a permutation : U U such that C i (u) = D i ( (u)) if C i (u) or D i ( (u)) is compromised or is adjacent to a compromised router. Then for every fair, cryptographic execution of C there exists an indistinguishable fair, cryptographic execution of D. The converse also holds. u v 12 3 4 5

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Relationship anonymity Corollary : A user and destination have rel. anonymity iff:

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Relationship anonymity 2 3 u 4? 5? The last router is unknown. Corollary : A user and destination have rel. anonymity iff:

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OR Relationship anonymity 2 3 u 4? 5? The last router is unknown. 1 2 4 The user is unknown and another unknown user has an unknown destination. 5 2? 5? 4? Corollary : A user and destination have rel. anonymity iff:

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OR 1 2 4 The user is unknown and another unknown user has a different destination. 5 1 2 Relationship anonymity 2 3 u 4? 5? The last router is unknown. 1 2 4 The user is unknown and another unknown user has an unknown destination. 5 2? 5? 4? Corollary : A user and destination have rel. anonymity iff:

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Probabilistic Anonymity

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Adding probability 1.To configurations a.User u selects destination d with probability p u d. b.Users select routers randomly. 2.To executions a.Each execution in a configuration is equally likely.

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Definition (relationship anonymity): u and d have relationship anonymity in configuration C with respect to adversary A if, for every fair, cryptographic execution of C in which u talks to d, there exists a fair, cryptographic execution that is indistinguishable to A in which u does not talk to d.

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Probabilistic relationship anonymity metric: Let X be a random configuration. Let X D : U D be the destinations in X. The metric Y for the relationship anonymity of u and d in C is: Y(C) = Pr[X D (u)=d | X C]

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Probabilistic Anonymity Other measures (e.g. entropy, min entropy) Measures effect on the individual user Exact Bayesian inference –Adversary after long-term intersection attack –Worst-case adversary

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Probabilistic Anonymity 1. Fixing u and d doesnt determine C. 2. Y(C) thus has a distribution. 3.This distribution depends on each p v. 4. E[Y | X D (u)=d] a.Worst case b.Typical case

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Worst-case Anonymity Theorem 2: The maximum of E[Y | X D (u)=d] over (p v ) v u occurs when 1. p v =1 for all v u OR 2. p v d =1 for all v u Let p u 1 p u 2 p u d-1 p u d+1 … p u

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Worst-case Estimates Let n be the number of nodes. Let b be the fraction of compromised nodes.

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Worst-case Estimates Theorem 3: When p v =1 for all v u: E[Y | X D (u)=d] = b + b(1-b)p u d + (1-b) 2 p u d [ (1-b)/(1-(1- p u ) b )) + O( logn/n) ] Let n be the number of nodes. Let b be the fraction of compromised nodes.

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Worst-case Estimates Theorem 3: When p v =1 for all v u: E[Y | X D (u)=d] = b + b(1-b)p u d + (1-b) 2 p u d [ (1-b)/(1-(1- p u ) b )) + O( logn/n) ] Theorem 4: When p v d =1 for all v u: E[Y | X D (u)=d] = b 2 + b(1-b)p u d + (1-b) p u d /(1-(1- p u d ) b ) + O( logn/n) ] Let n be the number of nodes. Let b be the fraction of compromised nodes.

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Worst-case Estimates Theorem 3: When p v =1 for all v u: E[Y | X D (u)=d] = b + b(1-b)p u d + (1-b) 2 p u d [ (1-b)/(1-(1- p u ) b )) + O( logn/n) ] Let n be the number of nodes. Let b be the fraction of compromised nodes.

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Worst-case Estimates Theorem 3: When p v =1 for all v u: E[Y | X D (u)=d] = b + b(1-b)p u d + (1-b) 2 p u d [ (1-b)/(1-(1- p u ) b )) + O( logn/n) ] b + (1-b) p u d Let n be the number of nodes. Let b be the fraction of compromised nodes.

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Worst-case Estimates Theorem 3: When p v =1 for all v u: E[Y | X D (u)=d] = b + b(1-b)p u d + (1-b) 2 p u d [ (1-b)/(1-(1- p u ) b )) + O( logn/n) ] b + (1-b) p u d E[Y | X D (u)=d] b 2 + (1-b 2 ) p u d Let n be the number of nodes. Let b be the fraction of compromised nodes.

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Typical Case Let each user select from the Zipfian distribution: p d i = 1/( i s ) Theorem 5: E[Y | X D (u)=d] = b 2 + (1 b 2 )p u d + O(1/n)

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Typical Case Let each user select from the Zipfian distribution: p d i = 1/( i s ) Theorem 5: E[Y | X D (u)=d] = b 2 + (1 b 2 )p u d + O(1/n) E[Y | X D (u)=d] b 2 + ( 1 b 2 )p u d

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Results 1.Formally model onion routing using input/output automata 2.Analyze relationship anonymity a.Characterize situations with possibilistic anonymity b.Bound probabilistic anonymity in worst-case and typical situations

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Future Work 1.Extend analysis to other types of anonymity and to other systems. 2.Examine how quickly users distribution are learned. 3.Analyze timing attacks.

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